In a certain crystalline material the vacancy concentration at 25 c is one-fourth that at 80 c. at what temperature would the vacancy concentration be 3 times that at 80 c?

At 93 °C, the vacancy concentration will be three times that at 80°C.

The formula for the vacancy concentration in a crystal is a form of the Arrhenius equation.

In logarithmic form, the equation is

ln(N_2/N_1) = (-Q/R)(1/T_2-1/T_1)

where

•Q = the energy required for vacancy formation

•N_2 = the vacancy concentration at T_2

•N_1 = the vacancy concentration at T_1

•R = the gas constant [8.314 J·K^(-1)mol^(-1)]

Let N_80 represent the vacancy concentration at 80 °C.

At 25 °C, ln(N_25/N_80) = ln(0.25N_80/N_80) = ln0.25 = -1.386

∴ -1.386 =(-Q/R)(1/298.15 – 1/353.15) = -1.306 × 10^(-4) × (Q/R)

Q/R = (-1.386)/[-1.306 × 10^(-4)] = 10 620

At T_2, ln(N_T2/N_80) = ln[(3N_80)/N_80] = ln3 = 1.099

∴ 1.099 = -10 620(1/T_2 – 1/353.15) = -10 620/T_2 + 10 620/353.15

= -10 620/T_2  + 30.072

10 620/T_2 = 30.072 – 1.099 = 28.97

T_2 = 10 620/28.97 = 366.4 K = 93 °C

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